- 01-434 Ostap Hryniv and Dima Ioffe
- Self-avoiding polygons:
Sharp asymptotics of canonical partition functions under the fixed
area constraint
(726K, ps)
Nov 26, 01
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Abstract. We study the self-avoiding polygons (SAP) connecting the
vertical and the horizontal semi-axes of the
positive quadrant of $\mathbb{Z}^2$. For a fixed $\beta>0$, assign to
each such polygon $\omega$ the weight $\exp\{-\beta|\omega|\}$,
$|\omega|$
denoting the length of $\omega$, and consider
the sum $Z_{Q,+}$ of these weights for all SAP enclosing area
$Q>0$. We study the statistical properties of such SAP and, in
particular, derive the exact asymptotics for the partition function
$Z_{Q,+}$ as $Q\to\infty$.
The results are valid for any $\beta >\beta_c$, $\beta_c$ being the
critical value for the 2D self-avoiding walks.
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